Saturday, June 2, 2012

I WENT TO COLLEGE IN BOSTON. WELL, NOT IN BOSTON, NEAR BOSTON. NO, NOT TUFTS:(A Guest Post from commenter J. Bowman) A while back, an assignment appeared in my inbox (stemming from this post, I believe), in which I was asked the following question:

Math 55, for those unwilling to click on the links, is the highest first-year class offered by the second-best math department in Cambridge, MA. The class is touted as being four years' worth of undergraduate mathematics (and fairly advanced undergraduate mathematics at that) in a single year, which leaves future mathematicians (and the occasional physicist or economist) free to pursue more specialized interests. Each year, 50-100 hopefuls sign up for the class; usually, about 20 make it to the end (most simply drop back to the somewhat less work-intensive Math 25).

Why was I asked to answer this question? For starters, I'm an actual mathematician; at least, that's what my business card says (though I am beginning to suspect that mathematician is an ancient Greek word for “C programmer”). My undergraduate degree comes from the other school in Cambridge, I am addressed as “Doctor” at work, and I do still occasionally fool around with some of the nastier bits of algebra and graph theory, both at Veridian and in my spare time (though my performance in ALOTT5MAball is an indication I should probably be focusing more on probability and statistics). So, in between weaponizing pumpkins and reading old XKCD strips, I took a gander at this year's version of Math 55. So, how hard is it?

The short answer: pretty darn hard. The long answer is the same, but I get to type more.
I looked through a couple of the problem sets, and they all had one thing in common: length. It is not uncommon for students to spend almost 60 hours a week on homework for this class, and solutions tend to be 15-20 (LaTeX-formatted) pages long. Rather than slog through those, I printed out the final exams, and brought them in to work on whenever we had to install new motion sensors in the labs (which happens with surprising frequency). I also passed them around to my colleagues, except for the guy who actually took Math 55; I figured his opinion would be biased.

The consensus on the first semester final was that it would make a fine graduate-level qualifying exam, especially if your school considered linear algebra worthy of inclusion on a qual (mine did, and it wasn't nearly this hard). The first question, using Peano's axioms to prove a cancellation property, is deceptively difficult; it appears you've practically been handed the answer (pssst... it's induction!), until you realize you'll have to derive the concept of less-than in order to get the last step. Frankly, the most frustrating thing about graduate-level algebra is the beginning of each sequence, in which you have to prove things that the average third-grader has seen and most middle schoolers simply know, and your professor won't give “Come ON! I knew how to subtract before I could tie my own shoes, this is obvious!” full credit (yes, I have tried that). Question five, on field extensions, made me chuckle, because I remembered doing that exact problem on a homework assignment—during my fourth year of grad school. The last question, though, is a thing of beauty, combining complex numbers and Galois theory to prove the solvability of a specific quintic polynomial. My favorite class ever spent an entire semester building the theory necessary to show that these polynomials are not, in general, solvable. This class did it in about three weeks.

The consensus on the second final could be summed up as, “Ugh.” All good-hearted mathematicians despise analysis. For the lay-people: analysis is best described as the “theory of calculus,” and not even mathematicians like calculus. Also, it includes differential equations, which is really physics in disguise. Basically, in the meatloaf of mathematics, analysis is the onions; the whole thing really doesn't work without it, but that doesn't mean I have to like it on its own.

Anyway, the problems are actually pretty standard stuff (though I did have to look up “isoperimetric”), but none of them are anything I would classify as easy, and the sheer diversity (this covers stuff I learned over three semesters) and volume would have had the freshman version of me writing long angry missives to my graders well before finishing. My favorite is problem 9, which ends with a calculation of a specific value of the Riemann-Zeta function. I've had to do this in at least two classes (diffEq and combinatorics), and it's always fun to see another way to get these values. It's also a good question because the value is fairly well-known, so you can figure out pretty quickly if you've made a mistake.

So how do I think I would have done? I have seen nearly all of this material before, and feel like I could certainly have handled this course with just a little refresher. But I have a PhD already, and if I was doing this, I wouldn't be doing anything else. The people taking this class are 18 years old, and are taking three or four other classes (though those are usually less intense). The 18-year-old version of me couldn't have gotten near this class; I probably would have ended up in Math 23, as I liked the math, but thought at the time that I was going to do something else. Another thing to keep in mind is that in the last couple of years, there has been a pretty straightforward treatment of the class, a semester of algebra followed by a semester of analysis. Previous years included things like point-set topology and differential geometry, which can be mind-blowing to experienced students.

There may be a couple of high-school math wizards reading this; if you even think you're up for the challenge of this class, then I say go for it. (as an aside: you should apply to Harvard whether or not you care about this class, or even going to Big H; if you get in, you can say you got in, and if you don't, they write the best rejection letters.), Know what you're in for, and realize that finishing this class truly is a special achievement, so not doing so won't close any doors for you. No matter what, if you even find this interesting, then bless you, because the world needs lots of people who find math interesting. Godspeed, and I look forward to hiring you for the Jabberwocky project someday.

28 comments:

This is such an interesting post -- thank you so much for writing it! It's hard for some of us English major types to even imagine what comes after calculus. And as someone who is currently homeschooling (math only) a mathy spaceboy, this makes me extremely glad to be handing him over to a real teacher in the fall.

On a related note, is it ok if I email you at some point and ask you approximately one million questions about becoming a mathematician?

I hovered over the link to verify you indeed were talking about that safety school down the river. Just wanted to let you know not to put it like this with a Lesley grad in the room, they have fearsome right hooks.

I subscribed to the Barbie theory of math early on and, in my post-schooling days, came to regret it. Sometimes I wonder what I could have done with it if I'd given it the same amount of attention I gave literature and the social sciences.

What comes after calculus is somebody sitting around thinking, "How can we make this harder?"Calculus, I think, is really the top of the heap in terms of both the discipline and attention to detail required to perform the computations, and the intuition necessary to understand why you're doing what you're doing, so that you can solve the next problem. Disciplines beyond that generally either move into ever-more-complex computation (DiffEq, finite element theory, linear algebra), or more layers of theory and abstraction (modern algebra, topology), but not both.

Anyway, feel free to email. I trust that you have access to someone who has access to the blog email account, and thus my address, so I don't have to type it here?

As a side note: If you head over to the Harvard Math homepage (the "second-best math department" link), you'll see that Evan O'Dorney was on Harvard's Putnam-winning team this year. The Putnam is a big math competition for undergrads. There are twelve questions, each worth 10 points, 3-4000 people take it each year (I never did), and the median score is usually 1 point. It's difficult.O'Dorney was, in fact, a Putnam Fellow, which means he was one of the top five scores in the competition. This is the sort of guy who makes me feel like a fraud for telling people I have a PhD in math.

This was awesome to read. I know nothing about math beyond the high school level -- ironically, the day I got my acceptance letter from the school being discussed in this post, I happily told my calculus teacher later that day that I would no longer spend any time stressing about his class, as Harvard didn't have a math requirement. But I did like math until that year, and I have always wondered about Math 55 as a conceptual matter. So, again: this was awesome to read!

Kim, your experience and mine were similar, albeit with different acceptance letters. Having placed out of any college math requirement, I managed to avoid math and math-related courses for four years, even as an undergrad business major, with a handful of exceptions in the core curriculum (Stat101, Econ 1 and 2, Fin101 and 102). I regret that more than a bit now, and I realize there's no way I'll remember enough calculus to help my kids if they get there in high school. But I still enjoy helping them now, and I'm looking forward to working through those challenges with them over the coming years. Once a mathlete, always a mathlete, I guess.

So here's a question: I feel like I am far from the only person in the world who sailed along in math before being generally befuddled by calculus. (For background - full on mathlete, did ok in calculus, but only because of a friend who taught me the "how tos" before every test, which I promptly forgot thereafter.) I have heard similar stories many times. Is this because (a) calculus is hard, yo, or is it possible that (b) calculus is sufficiently harder than "regular" math that your average high school math teacher isn't necessarily capable of bringing the degree of teaching skill required to get the students fully onboard? Or alternatively (c), this wasn't others' experience and my own mathematical abilities were more limited than they seemed to be during grades K-11. (Maybe this should be a post rather than a comment, but it's a little more QDay than ALOTT5MA.)

I don't remember being generally befuddled by calculus (notwithstanding the near lack of recall I have for it today), but I do remember having to work a LOT harder in AB and BC than I had in any previous math course. Now, I did have a very good teacher (at least that was my perception - it was a fairly demanding private school and he's still there 20+ years later, last I checked, so he's likely doing something right). So I think there's a whole lot of (a) calculus is hard, yo, at play, though (b) probably has a seat at the table in specific circumstances.

I sailed along in math with one exception until I was generally befuddled by calculus. Geometry was the exception, and over time I started to think that the link between that and calculus was spatial thinking and spatial visualization, which was my mathematical weakness. Geology strengthened that belief, as I had no trouble in that class until it came time to make drawings showing what was under the earth based on what we saw on the surface.

This may be my particular idiosyncrasy, or it may be why some people struggle in calculus. I could not for the life of me do things like visualize spheres within other three-dimensional shapes and calculate the volume between the two (I think that's what we did in AB or BC that killed me - eventually I dropped back to AB and was happier). I made it through Geometry with a very hard-fought mix of Bs and As, but BC Calculus was my Waterloo.

I was very relieved that my school didn't have distribution requirements so I didn't have to take any post-calculus math, though I did take Statistics and econ.

I'm right there with you. I was all about math in junior high; I loved all those Girls in Math programs and competitions and whatnot. And then there was calculus, which was when I decided to be a lawyer.

Genevieve & spacewoman, I am right there with you. I was fine in BC Calc - for some reason the visualizations there didn't give me a hard time - but then freshman year in college (at Isaac's alma mater, BOO HARVARD) I made the mistake of taking Astro 220, which was Astronomy with actual math involved rather than Astronomy for Poets or whatever they called the easy one that my friends with better foresight took. I just could not for the life of me deal with graphing in three dimensions - I went to every single office hours my Prof held (yes, I'm that girl...) and just COULD NOT GET IT. It was infuriating to hit up against something my brain just could not comprehend, but there it was. My incredibly kind (super caustic in lectures but then really nice when I came into his office week after week, confidence completely shot, to explain that I yet again did not understand the problem set) Prof basically took pity on me, gave me hints to the answers on the final exam (ethical? probably not. but he could tell how hard I was freaking trying...), and I ended up with a B+. It was both the worst and best class I took in 4 years of college, if only because it gave me some much-needed humility. YMMV on the aforementioned ethics of the Professor's pity on me and whether or not me being the only girl (seriously) in the class had anything to do with it. Frankly, I think he just didn't know what to do with someone who absolutely could not get it on her own.

It's not you; calculus is that hard. I posted something similar above, but in terms of both abstraction and complexity, calculus is a big leap over algebra (which is, itself, a bit of a jump from arithmetic). There are so many concepts that most students won't have seen before: real numbers, limits, epsilon-delta (which tends to also be the introduction to mathematical proof), continuity, combination and composition of functions... and that's before we even get to the derivative. By the time you reach the Mean Value Theorem and the Fundamental Theorem (my favorite theorem, despite my distaste for the discipline), most students have, as spacewoman puts it, decided to be a lawyer.

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